For a source with a given numberqof messages and an unspecified set of probabilities, the numberX(q)of non-trivially different compact codes that are possible increases in a predictable fashion asqincreases. Distinct binary compact codes ofqmessages correspond to distinct oriented binary trees withqterminal nodes. The theorem of this correspondence shows that, by using a recursion relation, and given that there is one compact code tree forq = 2, all compact code trees for anyq >2can be automatically constructed. This is done by splitting, for all integerss \geq 1, sbottom level nodes of all compact code trees which haveq - sterminal nodes and which end insor more bottom level nodes.